Q137 / 240 · KCET 2026 · Mathematics
The probability that a man and his wife live after $20$ years are $\dfrac{1}{4}$ and $\dfrac{1}{3}$ respectively. The probability that neither the man nor his wife live after $20$ years is
- A$\dfrac{3}{4}$
- B$\dfrac{1}{12}$
- C$\dfrac{7}{12}$
- Correct answerD$\dfrac{1}{2}$
Solution
Let $P(M)$ be the probability that the man lives after $20$ years and $P(W)$ be the probability that his wife lives after $20$ years.
Given $P(M) = \dfrac{1}{4}$ and $P(W) = \dfrac{1}{3}$.
The probability that the man does not live after $20$ years is $P(\overline{M}) = 1 - P(M) = 1 - \dfrac{1}{4} = \dfrac{3}{4}$.
The probability that the wife does not live after $20$ years is $P(\overline{W}) = 1 - P(W) = 1 - \dfrac{1}{3} = \dfrac{2}{3}$.
Assuming the events are independent, the probability that neither of them lives after $20$ years is $P(\overline{M} \cap \overline{W}) = P(\overline{M}) \times P(\overline{W})$.
$P(\overline{M} \cap \overline{W}) = \dfrac{3}{4} \times \dfrac{2}{3} = \dfrac{1}{2}$.
Answer: $\dfrac{1}{2}$
Given $P(M) = \dfrac{1}{4}$ and $P(W) = \dfrac{1}{3}$.
The probability that the man does not live after $20$ years is $P(\overline{M}) = 1 - P(M) = 1 - \dfrac{1}{4} = \dfrac{3}{4}$.
The probability that the wife does not live after $20$ years is $P(\overline{W}) = 1 - P(W) = 1 - \dfrac{1}{3} = \dfrac{2}{3}$.
Assuming the events are independent, the probability that neither of them lives after $20$ years is $P(\overline{M} \cap \overline{W}) = P(\overline{M}) \times P(\overline{W})$.
$P(\overline{M} \cap \overline{W}) = \dfrac{3}{4} \times \dfrac{2}{3} = \dfrac{1}{2}$.
Answer: $\dfrac{1}{2}$
Hence, the correct answer is option: 'D'.